Portfolios and CAPM: Alphas and Betas


Kerry Back

BUSI 721, Fall 2022
JGSB, Rice University

An important regression in finance is the regression of the excess return of an asset or portfolio or fund on the excess return of a benchmark.

Excess return = return minus risk-free rate

\[r-r_{f}=\alpha + \beta(r_b-r_f)+\epsilon\]

\(r\)=return
\(r_{f}\)=risk-free return
\(r_{b}\)=benchmark return
\(\epsilon\)=zero-mean risk uncorrelated with \(r_{b}\)

For example, asset = stock and benchmark = stock market return

Meaning of \(\beta\)

Beta answers this question:

if the benchmark is up 1%, how much do we expect the asset to be up, all else equal?

  • If \(\beta\)=2, we expect the asset to be up 2%
  • If \(\beta\)=0.5, we expect the asset to be up 0.5%

4A

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Meaning of \(\alpha\)

Alpha answers this question:

If I start by holding the benchmark, can I improve mean-variance efficiency by investing something in the asset?

  • The answer is “yes” if and only if \(\alpha\) > 0
  • If \(\alpha\) < 0, mean-variance efficiency can be improved by shorting the asset.

4B

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What Sharpe Ratio is Needed for > 0?

Performance is often measured by Sharpe ratio
= reward to risk ratio
= risk premium / std dev

\(\alpha > 0\) if and only if
Sharpe ratio > Sharpe ratio of benchmark \(\times\) correlation

Low correlation \(\rightarrow\) \(\alpha > 0\) with low Sharpe ratio.

4C

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